Some Problems in Stochastic Flows and Diffusions

这项研究包括两个领域的随机流和耦合扩散的问题。 在后一领域的主要问题涉及莫泽的Harnack不等式的概率证明和搜索流形上的最佳耦合。 在前一个领域，问题涉及布朗流和湍流的定性和定量行为，特别是它们的色散特性。耦合问题的一般设置是流形（例如表面）上的稳态温度分布的行为和收敛到该稳态的速率。 随机流的工作范围相当广泛。 它目前涉及浮油或海洋表面高温物体的分散性质。 最终目标是应用以前和当前工作中的一些想法来解决太阳磁场能量的分形性质问题。

This research encompasses problems in the two areas of stochastic flows and coupling of diffusions. The principal problems in the latter area involve a probabilistic proof of Moser's Harnack inequality and a search for the best coupling on manifolds. In the former area, the problems concern the qualitative and quantitative behavior of Brownian and turbulent flows, especially their dispersion properties.The general setting of the set of problems on coupling is the behavior of steady state temperature distributions on manifolds (surfaces for example) and the rate of convergence to this steady state. The scope of the work on stochastic flows is quite broad. It presently concerns the nature of dispersion of oil slicks or high temperature bodies on the ocean's surface. The ultimate goal is to apply some of the ideas from previous and current work to solve problems about the fractal nature of the energy of the magnetic field on the sun.